Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

Dispersion characteristics of water- and gold-infiltrated opal photonic crystals

This paper presents a theoretical analysis of conditions for the propagation of electromagnetic waves in a wide frequency range (from the infrared to ultraviolet spectral region) in synthetic opals infiltrated with water and gold nanoparticles. A dispersion equation is derived which describes the dispersion law of both “right-hand” (right-hand system of the [(E)\vec]\vec E, [(H)\vec]\vec H, and [(k)\vec]\vec k vectors) and “left-hand” (left-hand system of the [(E)\vec]\vec E, [(H)\vec]\vec H, and [(k)\vec]\vec k vectors) electromagnetic waves in the crystals. We have determined the dispersion characteristics of the refractive index and broadband reflectance of the opals, group velocity dispersion, and effective mass dispersion for phonons and polaritons. Theoretical results are compared to measured reflection spectra.     

The effective T-stress estimation and crack paths emanating from U-notches

The concept of the T-stress as a local constraint factor has been extended to U-notch tip stress distribution as the effective T-stress. The effective T-stress has been estimated as the average value of the T-stress in the region corresponding to the effective (characteristic) distance ahead of the notch tip. The T-stress is evaluated by finite element method using the experimental load for crack initiation and computing the difference between principal stresses along ligament. A large range of critical effective T-stress values is investigated for different specimen configurations and notch aspect ratios. Crack stabilisation and crack bifurcation for fracture emanating from notches according to the critical effective T-stress is discussed. A model involving the influence of the critical effective T-stress on void growth for ductile failure in the vicinity of the notch tip has been proposed.     

Kinetics of flow stress in ultra-pure tantalum single crystals in stress/temperature regime III

The temperature dependence of the critical resolved shear stress (CRSS), τ, of ultra-pure tantalum single crystals (RRR ≥ 14000) observed below 250 K for a range of shear-strain rates [(g)\dot] = 2×10 ^{- 5} - 6×10 ^{- 3}  \texts ^{- 1} \dot{\gamma } = 2\times 10^{ - 5} - 6\times 10^{ - 3} \,{\text{s}}^{ - 1} was analyzed within the framework of a kink-pair nucleation model of flow stress. The CRSS/strain-rate data follow the model formulation t ^{1/ 2} = C + D  ln[(g)\dot] \tau^{ 1/ 2} = C + D\,{ \ln }\dot{\gamma } , where C and D are positive constants, for each deformation temperature T in the range 78–250 K. Evaluation of the various slip-parameters of flow stress points to (211)[[`1]11] [\bar{1}11] slip system responsible for the yielding of ultra-pure tantalum single crystals in the so-called stress/temperature regime III (T < 250 K). The value of the pre-exponential factor [(g)\dot]_\texto \dot{\gamma }_{\text{o}} in the Arrhenius-type equation for the shear-strain rate [(g)\dot] \dot{\gamma } is found to be of the order of 10⁵ s⁻¹, which is substantially lower than that ( [(g)\dot]_\texto ~ 10⁷  \texts ^{- 1} ) \left( {\dot{\gamma }_{\text{o}} \sim 10^{7} \,{\text{s}}^{ - 1} } \right) determined in the stress/temperature regime II (250–400 K) and contradicts the assumption invariably made in most of the theoretical models of flow stress that [(g)\dot]_\texto \dot{\gamma }_{\text{o}} is a constant over a wide temperature range.     

Stress intensity factor K and the elastic T-stress for corner cracks

The stress intensity factor K and the elastic T-stress for corner cracks have been determined using domain integral and interaction integral techniques. Both quarter-circular and tunnelled corner cracks have been considered. The results show that the stress intensity factor K maintains a minimum value at the mid-plane where the T-stress reaches its maximum, though negative, value in all cases. For quarter-circular corner cracks, the K solution agrees very well with Pickard's (1986) solution. Rapid loss of crack-front constraint near the free surfaces seems to be more evident as the crack grows deeper, although variation of the T-stress at the mid-plane remains small. Both K and T solutions are very sensitive to the crack front shape and crack tunnelling can substantially modify the K and T solutions. Values of the stress intensity factor K are raised along the crack front due to crack tunnelling, particularly for deep cracks. On the other hand, the difference in the T-stress near the free surfaces and at the mid-plane increases significantly with the increase of crack tunnelling. These results seem to be able to explain the well-observed experimental phenomena, such as the discrepancies of fatigue crack growth rate between CN (corner notch) and CT (compact tension) test pieces, and crack tunnelling in CN specimens under predominantly sustained load.     

T-Stress in Elastic-plastic Crack-tip Fields

A ligament field ahead of a crack tip for Model-I plane strain problem in a power-law hardening material is developed. Based on this solution, a formula for the critical value of J-integral in terms of T-stress for an elastic-plastic material is derived. T-stress effects on fracture toughness are discussed.     

Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

Shear-thickening behaviour of concentrated polymer dispersions under steady and oscillatory shear

The rheological behaviour of a 58 vol.% dispersion of styrene/acrylate particles in ethylene glycol was investigated using a plate-on-plate rheometer. Experimental results showed that the concentrated polymer dispersion exhibited a strong shear-thickening transition under both steady shear and dynamic oscillatory conditions. The low-frequency dynamic oscillatory behaviour could be reasonably interpreted in terms of the steady shear behaviour. Accordingly, the critical dynamic shear rate [(g)\dot]_{\textc_d} , \dot{\gamma }_{{{\text{c\_d}}}} , agreed well with the critical shear rate obtained in steady flow [(g)\dot]_{\textc_s} , \dot{\gamma }_{{{\text{c\_s}}}} , where [(g)\dot]_{\textc_d} \dot{\gamma }_{{{\text{c\_d}}}} was calculated as the maximum shear rate by the critical dynamic shear strain γ _c and the frequency ω, i.e. [(g)\dot]_{\textc_d} = wg_\textc . \dot{\gamma }_{{{\text{c\_d}}}} = \omega \gamma_{\text{c}} . However, during high-frequency dynamic oscillation, it was observed that the shear thickening occurred only when an apparent critical shear strain was reached, which could not be fully explained by the wall-slipping effect. Based on freeze fracture microscopic observations, the effect of the micro-sized flocculation of particles on the rheology of concentrated dispersions was also discussed.     

Effect of the ratio of differently charged ions on the integral parameters of stationary plasma thrusters of the ATON type

The main integral parameters of stationary plasma thrusters (SPTs) of the ATON type with the outer channel diameters 60 mm (model A-3) and 100 mm (model A-5) are presented. The SPT characteristics have been studied for the propellant (Xe) consumption rates [(m)\dot]\dot m = 2 and 1 mg/s for A-3 and [(m)\dot]\dot m = 4 and 1.5 mg/s for A-5. Special attention has been devoted to the operation of SPT models at high voltages and low propellant consumption rates. An increase in the voltage has to be accompanied by a corresponding change in the magnetic field topograph and by an increase in the magnetic field intensity at the channel edge, which leads to a decrease in the discharge current and its oscillations. Model A-3 operating at [(m)\dot]\dot m = 1 mg/s and a voltage somewhat below 1000 V exhibited a specific momentum of ∼3400 s. The integral characteristics of SPTs depend on the ratio of differently charged ions in the plasma. The most efficient operation of the plasma source is observed in the case of equal concentrations of singly and doubly charged xenon ions. A new method is proposed for evaluation of the limiting SPT efficiency using only the current-voltage and propulsion characteristics of discharge.     

Cracked asphalt pavement under traffic loading – A 3D finite element analysis

An asphalt pavement containing a transverse top-down crack is investigated under traffic loading using 3D finite element analysis. The stress intensity factors (SIFs) and the T-stress are calculated for different distances between the crack and the vehicle wheels. It is found that all the three Modes (I, II and III) are present in the crack deformation. The signs and magnitudes of K_I, K_II, K_III and T are significantly dependent on the location of the vehicle wheels with respect to the crack plane. The magnitude of T-stress is considerable, if compared to the stress intensity factors, when one of the wheels is very close to the crack plane.     

Two-parameter fracture criterion (Kρ,c-Tef,c) based on notch fracture mechanics

A two-parameter fracture criterion has been proposed to predict fracture conditions of notched components. This criterion includes the critical notch stress intensity factor K _ρ,c , which represents fracture toughness of a material with a notch of radius ρ, and the effective T-stress. The effective T-stress T _ef has been estimated as the average value of the T-stress distribution in the region ahead of the notch tip at the effective distance X _ef . These parameters were derived from the volumetric method of notch fracture mechanics. The results of numerical T _ef,c -stress estimation are compared to the T _ef,c -stress results obtained from experimental analysis. The material failure curve or master curve K _ρ,c = f(T _ef,c ) has been established as a result of the notched specimen tests. A large T _ef,c range was covered from −0.80 σ _Y to +0.19 σ _Y using SENT, CT, RT (roman tile) and DCB specimens. It was shown that the notch fracture toughness is a linear decreasing function of the T _ef,c -stress. The use of the material failure curve to predict fracture conditions was demonstrated on gas pipes with the surface notch.     

Application of a conservation integral to an interface crack interacting with singularities

A mutual integral, which has the conservation property is applied to the problem of an interfacial crack. The stress intensity factors K ₁, K ₂, K ₃ and T-stress for the problem in an infinite medium are easily obtained by using the mutual integral without solving the boundary value problem. In doing so the auxiliary solutions are required and they are taken from the known asymptotic solutions. This method is amenable to numerical evaluation of the stress intensity factors and T-stress if the interfacial crack in a finite medium is considered.     

Numerical modelling of crack path in the lubricated rolling–sliding contact problems

A two-dimensional computational model for simulation of contact fatigue of lubricated rolling–sliding contact problems is presented. In the model it is assumed that the initial crack of length 0.02 mm is initiated at the surface due to previous mechanical or heat treatment of the material as well as a consequence of running process in an early stage of exploitation. The discretised model with the initial crack is then subjected to the normal contact pressure and tangential loading due to friction between the contacting surfaces. The model also considers the moving contact of the contacting surfaces and fluid trapped in the crack. The crack propagation path is predicted with the MTS and modified MTS criterion, which takes into account the influence of the stress intensity factor K_I and K_II, T-stress, stress on the crack surface caused by lubricant pressure inside the crack and critical distance ahead the crack tip. The numerical results correspond well with available experimental data.     

Maximum Average Tangential Stress Criterion for Prediction of the Crack Path

The concept of the average stress has been employed to propose the maximum average tangential stress (MATS) criterion for predicting the direction of fracture angle. This criterion states that a crack grows when the maximum average tangential stress in the fracture process zone ahead of the crack tip reaches its critical value and the crack growth direction coincides with the direction of the maximum average tangential stress along a constant radius around the crack tip. The tangential stress is described by the singular and non-singular (T-stress) terms in the Williams series solution. The predicted directions of fracture angle are consistent with the experimental data for the mixed mode I/II crack growth behavior of Guiting limestone.     

Numerical investigation of crack tip strain localization under cyclic loading in FCC single crystals

In this work, the crack tip strain localization in a face centered cubic single crystal subject to both monotonic and cyclic loading was investigated. The effect of constraint was implemented using T-stress and strain accumulation was studied for both isotropic and anisotropic elastic cases with the appropriate application of remote displacement fields in plane strain. Modified boundary layer simulations were performed using the crystal plasticity finite element framework. The consideration of elastic anisotropy amplified the effect of constraint level on stress and plastic strain fields near the crack tip indicating the importance of its use in fracture simulations. In addition, to understand the cyclic stress and strain behavior in the vicinity of the crack tip, combined isotropic and kinematic hardening laws were incorporated, and their effect on the evolution of yield curves and plastic strain accumulation were investigated. With zero-tension cyclic load, the evolution of plastic strain and Kirchhoff stress components showed differences in magnitudes between isotropic and anisotropic elastic cases. Furthermore, under cyclic loading, ratcheting was observed along the localized slip bands, which was shown to be affected by T-stress as well as elastic anisotropy. Negative T-stress increased the accumulation of plastic strain with number of cycles, which was further amplified in the case of elastic anisotropy. Finally, in all the cyclic loading simulations, the plastic strain accumulation was higher near the \(55^0 \) slip band.     

T-Stress Effects on Isochromatic Fringe Patterns in Mode II

The theory of photoelasticity is used to study analytically the effects of T-stress on the fringe patterns around the crack tip in mode II crack specimens. The locus of an isochromatic fringe determined by taking into account the T-stress is compared with the locus of a fringe with no T-stress. It is shown for mode II cracks that in the presence of T-stress, the fringe loops are neither symmetric nor continuous. Asymmetric and discontinuous fringe patterns predicted in this paper are consistent with the experimental results observed previously in photoelasticity tests.     

A brief note on elastic T-stress for centred crack in anisotropic plate

The stress intensity factors (SIFs) and the T-stress for a planar crack with anisotropic materials are evaluated by the fractal finite element method (FFEM). The FFEM combines an exterior finite element model and a localized inner model near the crack tip. The mesh geometry of the latter is self-similar in radial layers around the tip. A higher order displacement series derived from Laurent series and Goursat functions is used to condense the large numbers of nodal displacements at the inner model near the crack tip into a small set of unknown coefficients. In this study, the variations of the SIFs and the T-stress with material properties and orientations of a crack are presented. The separation of the analytical displacement series into four fundamental cases has shown to be necessary in order to cover all the material variations and the orientations of a crack in the plate with general rectilinear anisotropic materials.     

The in-plane and out-of-plane stress constraint factors and K-T-Tz description of stress fields near the border of a quarter-elliptical corner crack

The elastic T-stress and stress intensity factor K for quarter-elliptical corner cracks have been investigated in elastic plates by detailed three-dimensional finite-element calculations. The distributions of normalized K and T-stress have been obtained along the crack front with aspect ratios (a/c) of 0.2, 0.3, 0.4, 0.5, 0.6, 0.8 and 1.0, and far-field tension and the effect of Poisson's ratio have also been considered. The normalized K increases and the normalized T-stress decreases with the increase of Poisson's ratio v. For v= 0.3, the normalized K gradually increases in the range of crack-face angle φ≥ 22.5° and decreases in the range of φ≤ 22.5° with the increase of a/c. The normalized T-stress increases in the beginning and then decreases with increasing φ except for a/c= 0.2 and a/c= 0.3. By fitting the numerical results with the least squares method, empirical formulae have been given for the convenience of engineering applications. Combining with the corresponding out-of-plane constraint factor T_z, the three-parameter K-T-T_z approach has been provided, which can accurately describe the stress field around the crack front.     

Effect of lattice orientation on crack tip constraint in ductile single crystals

In this work, the effect of lattice orientation on the fields prevailing near a notch tip is investigated pertaining to various constraint levels in FCC single crystals. A modified boundary layer formulation is employed and numerical solutions under mode I, plane strain conditions are generated by assuming an elastic–perfectly plastic FCC single crystal. The analysis is carried out corresponding to different lattice orientations with respect to the notch line. It is found that the near‐tip deformation field, especially the development of kink or slip shear bands is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the notch tip are also strongly influenced by the level of T ‐stress. The present results clearly establish that ductile single crystal fracture geometries would progressively lose crack tip constraint as the T ‐stress becomes more negative irrespective of lattice orientation. Also, the near‐tip field for a range of constraint levels can be characterized by two‐parameters such as K–T or J–Q as in isotropic plastic solids.     

A fracture mechanics model for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems

A computational model is presented for the analysis of micro-pitting in regard to lubricated rolling–sliding contact problems. This model assumes the appearance of an initial microcrack on the contact surface due to the mechanical or thermal treatment of the material, and as a consequence of an on-going process in early the stage of exploitation. The discretised model of the contacting mechanical elements is subjected to normal loading (Hertzian contact pressure), tangential loading (friction between contacting surfaces) and internal pressure to the crack surfaces. Crack propagation is predicted as follows: (1) using modified maximum tangential stress criterion, which takes into account the influence of stress intensity factors K_I and K_II, T-stress, stress on the crack’s surface caused by lubricant pressure inside the crack, and the critical distance ahead of the crack tip and (2) the classical maximum tangential stress criterion, which only takes into account the influence of the stress intensity factors K_I and K_II. The stress intensity factor based on these two criteria is then used in a short crack growth theory to determine the fatigue life of an initial crack to extent up to micro-pit. The developed model is applied to a real spur gear pair.